Portfolios and CAPM: Alphas and Betas





Kerry Back

An important regression in finance is the regression of the excess return of an asset or portfolio or fund on the excess return of a benchmark.

Excess return = return minus risk-free rate

\[r-r_{f}=\alpha + \beta(r_b-r_f)+\epsilon\]

\(r\)=return
\(r_{f}\)=risk-free return
\(r_{b}\)=benchmark return
\(\epsilon\)=zero-mean risk uncorrelated with \(r_{b}\)

For example, asset = stock and benchmark = stock market return

Meaning of \(\beta\)

Beta answers this question:

if the benchmark is up 1%, how much do we expect the asset to be up, all else equal?

  • If \(\beta\)=2, we expect the asset to be up 2%
  • If \(\beta\)=0.5, we expect the asset to be up 0.5%

4A

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Meaning of \(\alpha\)

Alpha answers this question:

If I start by holding the benchmark, can I improve mean-variance efficiency by investing something in the asset?

  • The answer is “yes” if and only if \(\alpha\) > 0
  • If \(\alpha\) < 0, mean-variance efficiency can be improved by shorting the asset.

4B

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What Sharpe Ratio is Needed for > 0?

Performance is often measured by Sharpe ratio
= reward to risk ratio
= risk premium / std dev

\(\alpha > 0\) if and only if
Sharpe ratio > Sharpe ratio of benchmark \(\times\) correlation

Low correlation \(\rightarrow\) \(\alpha > 0\) with low Sharpe ratio.

4C

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